{"id":2733,"date":"2012-08-03T18:41:28","date_gmt":"2012-08-03T18:41:28","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2733"},"modified":"2012-08-03T18:41:28","modified_gmt":"2012-08-03T18:41:28","slug":"scheme-theoretically-dense","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2733","title":{"rendered":"Scheme theoretically dense"},"content":{"rendered":"<p>Let X be a scheme and let U be an open subscheme. The <em>scheme theoretic closure<\/em> of U in X is the smallest closed subscheme Z of X such that j : U &#8212;> X factors through Z. We say that U is <em>scheme theoretically dense<\/em> in X if the scheme theoretic closure of U &cap; V in V equals V for every open V of X. See <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/01RB\">Definition Tag 01RB<\/a>. Then U is scheme theoretically dense in X if and only if O_X &#8212;> j_*O_U to be injective, see <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/01RE\">Lemma Tag 01RE<\/a>.<\/p>\n<p>If X is locally Noetherian, then U is scheme theoretically dense in X if and only if U is dense in X and contains all embedded points of X (<a href=\"http:\/\/stacks.math.columbia.edu\/tag\/083P\">Lemma Tag 083P<\/a>).<\/p>\n<p>For general schemes the situation isn&#8217;t as nice. For example, there exists a scheme with 1 point but no associated point (<a href=\"http:\/\/stacks.math.columbia.edu\/tag\/05AI\">Lemma Tag 05AI<\/a>). As a replacement for associated points, we sometimes use weakly associated primes (<a href=\"http:\/\/stacks.math.columbia.edu\/tag\/0547\">Definition Tag 0547<\/a>) and the corresponding notion for schemes. This notion agrees with associated point for locally Noetherian schemes. There are enough weakly associated points: if U contains all the weakly associated points, then U is scheme theoretically dense (result not yet in the stacks project). But in some sense there are too many: there is an example of a scheme theoretically dense open subscheme U of a scheme X which does not contain all weakly associated points of X (<a href=\"http:\/\/stacks.math.columbia.edu\/tag\/084J\">Section Tag 084J<\/a>).<\/p>\n<p>We have the following result from Raynaud-Gruson: If X &#8212;> Y is an etale morphism and x &isin; X with image y &isin; Y then x is a weakly associated point of X if and only if y is a weakly associated point of Y (<a href=\"http:\/\/stacks.math.columbia.edu\/tag\/05FP\">Lemma Tag 05FP<\/a>).<\/p>\n<p>What about scheme theoretic density? Given an etale morphism of schemes g : X&#8217; &#8212;> X and a scheme theoretically dense open U &subset; X the inverse image g^{-1}U is a scheme theoretically dense in X&#8217; (<a href=\"http:\/\/stacks.math.columbia.edu\/tag\/0832\">Lemma Tag 0832<\/a>). This was added recently in order to show that scheme theoretic density defined as above (and as in EGA IV 11.10.2) makes sense in the setting of algebraic spaces.<\/p>\n<p>If you have trouble falling asleep tonight, try proving some of the results above.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let X be a scheme and let U be an open subscheme. The scheme theoretic closure of U in X is the smallest closed subscheme Z of X such that j : U &#8212;> X factors through Z. We say &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2733\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2733","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2733","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2733"}],"version-history":[{"count":9,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2733\/revisions"}],"predecessor-version":[{"id":2742,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2733\/revisions\/2742"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2733"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2733"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2733"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}